Answer
$$\eqalign{
& {\text{Trapezoidal Rule}} \approx 1.4298 \cr
& {\text{Simpson's Rule}} \approx 1.4582 \cr
& {\text{Graphing utility}} \approx 1.4578 \cr} $$
Work Step by Step
$$\eqalign{
& \int_{\pi /2}^\pi {\sqrt x \sin x} dx \cr
& {\text{*Using the trapezoidal Rule }}\left( {{\text{THEOREM 4}}{\text{.17}}} \right) \cr
& \int_a^b {f\left( x \right)} dx \approx \frac{{b - a}}{{2n}}\left[ {f\left( {{x_0}} \right) + 2f\left( {{x_1}} \right) + \cdots 2f\left( {{x_{n - 1}}} \right) + f\left( {{x_n}} \right)} \right] \cr
& {\text{For }}n = 4,{\text{ }}\Delta x = \frac{{b - a}}{n} = \frac{{\pi - \pi /2}}{4} = \frac{\pi }{8},{\text{ then,}} \cr
& \cr
& {x_0} = \frac{\pi }{2},{\text{ }}{x_1} = \frac{5}{8}\pi ,{\text{ }}{x_2}{\text{ = }}\frac{3}{4}\pi {\text{, }}{x_3} = \frac{7}{8}\pi ,{\text{ }}{x_4} = \pi \cr
& f\left( {{x_0}} \right) = f\left( {\frac{\pi }{2}} \right) = \sqrt {\frac{\pi }{2}} \sin \left( {\frac{\pi }{2}} \right) = \sqrt {\frac{\pi }{2}} \cr
& f\left( {{x_1}} \right) = f\left( {\frac{5}{8}\pi } \right) = \sqrt {\frac{5}{8}\pi } \sin \left( {\frac{5}{8}\pi } \right) \cr
& f\left( {{x_2}} \right) = f\left( {\frac{3}{4}\pi } \right) = \sqrt {\frac{3}{4}\pi } \sin \left( {\frac{3}{4}\pi } \right) = \frac{{\sqrt {6\pi } }}{4} \cr
& f\left( {{x_3}} \right) = f\left( {\frac{7}{8}\pi } \right) = \sqrt {\frac{7}{8}\pi } \sin \left( {\frac{\pi }{2}} \right) \cr
& f\left( {{x_4}} \right) = f\left( \pi \right) = \sqrt \pi \sin \left( \pi \right) = 0 \cr
& \cr
& \int_{\pi /2}^\pi {\sqrt x \sin x} dx \approx \frac{\pi }{{16}}\left[ {\sqrt {\frac{\pi }{2}} + 2\sqrt {\frac{5}{8}\pi } \sin \left( {\frac{5}{8}\pi } \right) + 2\left( {\frac{{\sqrt {6\pi } }}{4}} \right)} \right] \cr
& + \frac{\pi }{{16}}\left[ {2\sqrt {\frac{7}{8}\pi } \sin \left( {\frac{{7\pi }}{8}} \right) + 0} \right] \cr
& {\text{Simplifying by using a calculator}} \cr
& \int_{\pi /2}^\pi {\sqrt x \sin x} dx \approx 1.42986 \cr
& \cr
& {\text{*Using the Simpson's Rule }}\left( {{\text{THEOREM 4}}{\text{.19}}} \right) \cr
& \int_a^b {f\left( x \right)} dx \approx \frac{{b - a}}{{3n}}\left[ {f\left( {{x_0}} \right) + 4f\left( {{x_1}} \right) + 2f\left( {{x_2}} \right) + 4f\left( {{x_3}} \right) + \cdots } \right. \cr
& \left. { + 4f\left( {{x_{n - 1}}} \right) + f\left( {{x_n}} \right)} \right] \cr
& \int_{\pi /2}^\pi {\sqrt x \sin x} dx \approx \frac{\pi }{{24}}\left[ {\sqrt {\frac{\pi }{2}} + 4\sqrt {\frac{5}{8}\pi } \sin \left( {\frac{5}{8}\pi } \right) + 2\left( {\frac{{\sqrt {6\pi } }}{4}} \right)} \right] \cr
& {\text{ }} + \frac{\pi }{{24}}\left[ {4\sqrt {\frac{7}{8}\pi } \sin \left( {\frac{{7\pi }}{8}} \right) + 0} \right] \cr
& {\text{Simplifying by using a calculator}} \cr
& \int_{\pi /2}^\pi {\sqrt x \sin x} dx \approx 1.4582 \cr
& \cr
& {\text{Using a graphing utility we obtain}} \cr
& \int_{\pi /2}^\pi {\sqrt x \sin x} dx \approx 1.4578 \cr} $$
